Acta mathematica scientia, Series B >
MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP
Received date: 2007-08-16
Online published: 2009-09-20
Supported by
Sponsored by the NSFC (10871003, 10701008, 10726064), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007001040)
In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are
studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in $L^2({\mathbb H}^d)$ by using self-similar tilings for the acceptable dilations on the Heisenberg group.
Key words： Heisenberg group; multiresolution analysis; wavelets; self similar tilings
LIU He-Ping , LIU Yu , WANG Hai-Hui . MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP[J]. Acta mathematica scientia, Series B, 2009 , 29(5) : 1251 -1266 . DOI: 10.1016/S0252-9602(09)60102-8
[1] Folland G B. Harmonic Analysis in Phase Space. New Jersey: Princeton University Press, 1989
[2] Franchi B. Rectifiability and perimeter in the Heisenberg group. Math Ann, 2001, 321: 479--531
[3] Gelbrich G. Self-similar periodic tilings on the Heisenberg group. J Lie Theory, 1994, 4: 31--37
[4] Gröchenig K, Madych W R. Multiresolution analysis, Haar bases, and self-similar tilings of Rn. IEEE Transactions on Information Theory, 1992, 38: 556--568
[5] Jawerth B, Peng L Z. Compactly supported orthogonal wavelets on the Heisenberg group. Beijing: Research Report No.45 of Insititute Mathematics of Peking University, 2001
[6] Kigami J. Analysis on Fractals. Cambridge: Cambridge University Press, 2001
[7] Lawton W. Infinite convolution products & refinable distributions on Lie groups. Trans Amer Math Soc, 2000, 352: 2913--2936
[8] Liu H P, Peng L Z. Admissible wavelets associated with the Heisenberg group. Pacific J Math, 1997, 180: 101--123
[9] Liu H P, Liu Y, Peng L Z, et al. Cascade algorithm and multiresolution analysis on the Heisenberg
group. Progress in Natural Science, 2005, 15: 602--608
[10] Liu H P, Liu Y. Refinable functions on the Heisenberg group. Comm Pure Appl Anal, 2007, 6: 775--787
[11] Liu M J, Lu S Z. A weighted estimate of the Hormander multiplier on the Heisenberg group. Acta Math Sci, 2006, 26B(1): 134--144
[12] Liu Y, Yu M. Lipschitz continuity of refinable functions on the Heisenberg group. J Math Anal Appl, 2008, 338: 1081--1091
[13] Peng L Z. Wavelets on the Heisenberg Group. Geometry and Nonlinear Partial Differential Equations, AMS/IP Studies in Advanced Mathematics, 2002, 29: 123--131
[14] Stein E M. Harmonic Analysis Real-Variable Methods, Orthogonality, and Oscillatory Integrals. New Jersey: Princeton University Press, 1993
[15] Strichartz R S. Self-similarity on nilpotent Lie group. Contemporary Mathematics, 1992, 140: 123--157
[16] Strichartz R S. Wavelet and self-affine tilings. Constr Approx, 1993, 9: 327--346
[17] Yang Q. D. Multiresolution analysis on non-abelian locally compact groups
[D]. University of Saskatchewan, 1999
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